Area Problem Calculus

area problem calculus
surface area can calculus problem?

hi. i would like to find out how to solve this problem.

You have been asked to design a can shaped like a right circular cylinder with height h and radius r. Given that the can must hold exactly 810 cm^3, what values of h and r will minimise the total surface area (including the top and bottom faces)?

The answers should be correct to 2 decimal places as a list [in brackets] of the form: [ h, r ]
for constants h (height), r (radius), in that order.

cm^3 means cubic centimetres

thanks for any help. it is very much appreciated

810=(pi)r^2h

SA=2(pi)rh+2(pi)r^2

I’d take the derivative of the SA. This will give you either a min or a max. But first, I would try to solve SA to be in terms of just one variable.

Let’s now take a derivative of SA with respect to r.
d(SA)/dr= -(1620/r^2)+4(pi)r
When this equals zero, we have a min or a max.
0=-(1620/r^2)+4(pi)r
I’m going to multiply everything by r^2.
0=-1620+4(pi)r^3
1620=4(pi)r^3
[1620/(4pi)]^(1/3)=r=5.05
Using our original equation, we now solve for h:
810=(pi)r^2h
Answer:[10.10, 5.05]

Calculus Area Problem Part IV: The Conclusion of the Area Problem